|Before you lies a tower of champagne glasses that must be filled. Each glass rests upon exactly 2 other glasses beneath it, with the exception of the lowest row, which sits on the table. An example of a 5 glass tall tower is shown below:|
4 5 6
7 8 9 10
11 12 13 14 15
Each glass can hold 2 units of champagne, and if more than 2 units of champagne fall into any glass, all but 2 of the units will overflow equally into the two glasses below it. Thus, all of the champagne poured into glass 1 after the first 2 units will fall equally into glasses 2 and 3. Once 2 and 3 overflow they will each, independently, distribute the champagne into the two glasses beneath them. At that point glass 5 will be receiving champagne from both glasses 2 and 3. Glass 4 will be receiving champagne from just glass 2, and glass 6 will be receiving champagne from just glass 3.
Given the height of the tower, the number of units poured into glass 1, and the number of a particular glass, return what fraction of that glass is full. Your solution should be a reduced fraction (lowest terms) in the form "numerator/denominator" with no leading zeros in either the numerator or denominator. Full glasses are denoted by "1/1" whereas empty glasses are denoted "0/1". The glasses are numbered in the same way as above, namely left to right, top to bottom. Note, once a glass on the bottom layer is full, extra champagne will just spill onto the table.